Most students are first introduced to physics in the form of kinematics--the branch of physics that studies only the motion of objects. They use equations to calculate velocity, position and acceleration to learn how to apply mathematics to the real world. A common question asks students to calculate the final velocity of an object without knowing how long it accelerated. As long as the acceleration and displacement of the object is known, any student can solve this problem.

### Problem Analysis

Verify that acceleration is constant. Constant acceleration will be a simple number such as 9.8 meters per second per second (m/s^2), and will not change depending on time or velocity.

Examine the problem to find the displacement of the object and its initial velocity.

Plug the acceleration, displacement and initial velocity into this equation: (Final Velocity)^2 = (Initial Velocity) ^2 + 2_(Acceleration)_(Displacement). Solve the problem using pen, paper and calculator.

### Sample Problem

Suppose that a car has an initial velocity of 5 meters per second, and it accelerates at 4 meters per second per second over a distance of 10 meters. You can find its final velocity and how long the car took to travel 10 meters.

Identify the acceleration of the car. Here, it's clearly stated as 4 m/s^2. This is constant acceleration because it does not change over time; the car's acceleration is the same throughout the problem.

Find the initial velocity and displacement. The initial velocity is clearly stated as 5 meters per second. But the problem states only the distance traveled and not the displacement. Use intuition to assert that distance traveled and displacement are the same, 10 meters.

Solve the equation (Final Velocity)^2 = (Initial Velocity) ^2 + 2_(Acceleration)_(Displacement). Plugging values in gives V^2 = 5^2 + 2_4_10. Taking the square root of both sides (and using intuition again to assert that the result should be positive) gives V equals the square root of (25+80) = 10.25 meters per second.

Solve for time after final velocity is found. You may rearrange the following equation to do this: (Final Velocity) = (Initial Velocity) + (Acceleration)*(Time). So in this case, (Time) = (Final Velocity - Initial Velocity)/(Acceleration). Time then equals (10.25 - 5)/(4) = 1.31 seconds.

#### Tip

Simple algebra mistakes are the most common error students make in kinematics problems.